### February 2010

A departure from directly working with varieties, we’re going to do something that’s strictly topological (at first glance) but which really has deep and important connections with Hodge theory. We’re going to talk about monodromy and monodromy representations.

Let ${F\rightarrow E\rightarrow B}$ be a fiber bundle. Though we don’t really NEED any hypotheses, as far as I’m aware, because our real goal is the study of varieties, we’ll assume that the fiber, ${F}$ is a complete variety, and that the base, ${B}$, is a variety (often open). So really, we’re fiddling with families of varieties, but just topologically for the moment, so we’ll ignore any degenerations (those will come later).

In the meantime, monodromy is a trick to get some nice representations of ${\pi_1(B)}$. First, we’ll look at the motivating example for a lot of this:

\bf{Example}: Ignore most of the stuff above, and just look at ${p:\tilde{X}\rightarrow X}$ a covering space of degree ${n}$. Then there’s a map ${\pi_1(X)\rightarrow S_n}$ given by the action of ${\pi_1(X)}$ by interchanging the sheets of the cover.

So this gave us a map from the fundamental group of the base into the automorphism group of the fibers. This actually happens in general, for any fiber bundle, just go around the loop and see how the transition functions change. But that’s not the most interesting case (though it can be adapted into a nice proof that ${\mathcal{M}_g}$ is irreducible). The problem being that the transition maps may come from ridiculously huge groups, things like ${\mathrm{Diff}_+(S^1)}$, and while that’s a group I do rather like, it’s not so helpful for getting good representations.

But what tools can we possibly have to linearize something on a space? Homology and cohomology, of course! Here’s an example.

\bf{Example}: Look at the family ${y^2z=x(x-z)(x-\lambda z)}$ of projective elliptic curves. We’re ignoring degenerations for the moment (we’ll complete families later) so this lives over ${\mathbb{P}^1\setminus\{0,1,\infty\}}$. This is also visible as ${\mathbb{C}^\times\setminus\{1\}}$. So we’ve got two nontrivial loops, one around zero, the other around one, and there are no relations, this is the free group of rank 2. We can envision all of this as living in the plane with four points marked as ${0,1,\infty,\lambda}$, and a cut from ${0}$ to ${\lambda}$ and a cut from ${1}$ to ${\infty}$. Then the loops are what happens if we loop ${\lambda}$ around things. We get one loop from rotating the cut between ${0}$ and ${\lambda}$ all the way around. The other one is a bit trickier to describe directly, but we can describe it in terms like this.

We’ll focus on the first one. Look at ${H_1}$ of the elliptic curve. It has two generators, call them ${\delta}$ and ${\gamma}$, where ${\delta}$ is a loop around the ${0\lambda}$ cut, and ${\gamma}$ is a loop through the two cuts. This is a standard homology basis, and we’ll look at the action of our element of ${\pi_1(\mathbb{P}^1\setminus\{0,1,\infty\}})$ on the homology.

If we rotate the cut only 180 degrees, we get the square root of the correct matrix, so we’ll have to square whatever it is we get out of this, but the setup is easier to describe. Call out 180 degree turn operator ${T}$. Then it’s fairly easy to see that ${T(\delta)=\delta}$, because ${\delta}$ is just brought back to itself. However, ${T(\gamma)}$ is a bit more interesting. It will be ${T(\gamma)=a\delta+b\gamma}$, and we can compute ${a,b}$ by intersecting with ${\gamma}$ and ${\delta}$. Doing this gives ${T(\gamma)=\delta+\gamma}$, and so our matrix is ${\left(\begin{array}{cc}1&1\\ 0&1\end{array}\right)}$. Squaring it, we get the matrix ${\left(\begin{array}{cc}1&2\\ 0&1\end{array}\right)}$. If we do the same procedure for the other generator, we get ${\left(\begin{array}{cc}1&0\\2&1\end{array}\right)}$, which is a perfectly good representation of our free group, on ${2\times 2}$ integer valued matrices!

This sort of thing will work in general, though computing the representation is a bit trickier when the base of the family is higher dimensional, because we can’t have as nice pictures in our heads. But in particular, we get a representation of ${\pi_1}$ on the automorphism group, and on all of the homology (and thus, cohomology) spaces.

If you like Hyperkähler manifolds (and who doesn’t?) go check out Beauville’s new preprint: Holomorphic symplectic geometry: a problem list.  It’s nice and short survey of the basic facts of hyperkähler manifolds, including a bunch of conjectures and open problems in the area.

Previously, we talked a bit about the category of Hodge structures, and did some basic constructions.  However, I’d claimed that this was algebraic geometry (at least, in the categories on the post) so today, we’ll talk about a LOT of Hodge structure that arise in nature.  Everything I say is true in more generality (for compact Kähler manifolds, in fact) but for now, let $X$ be a smooth projective variety (as always, over $\mathbb{C}$.

Over at the n-Category Cafe, Tom Leinster has written an excellent post pointing out that sheaf theory is NOT a subfield of algebraic geometry.  I feel I have a few things to add, not enough for a long post, but enough that I’d rather post here than fill up their comment thread (plus, the blatant cry for traffic, of course, as I am trying to bring this blog back to life).

So first off, I agree completely, sheaves are not owned by AG.  In fact, I’d go so far as to say (in my completely reckless manner) that AG is a special case of the theory of sheaves, one where we can actually say a lot!

So first, what’s a sheaf? I’ve talked about this before, but let’s review quickly.  Take a category, any category, and give it a topology, pick your favorite one.  Then we define a sheaf of objects of a second category $\mathcal{C}$ to be a contravariant functor satisfying a gluing condition in the topology on your category, spelled out in detail in my old post.

So, what’s a scheme? It’s just a sheaf in the Zariski topology on CommRing satisfying some conditions! (what they are is unimportant) So the general study of sheaves is far, far more general than the schemes.  What about spaces? Well, same thing works! Stacks are a bit trickier, and I don’t understand them as well, but they’re also a purely categorical notion that is related to sheaf theory (some fuzziness due to 2-categories, I believe) and algebraic stacks are just particular ones satisfying some extra hypotheses.

So, I’ve said why algebraic geometry is owned by sheaves, but not what else is.  As far as I can see, it’s pretty much the entirety of mathematics.  Well…not all of it, but a ridiculously large swath.  Sadly, most people in the other areas are terrified of sheaves or think “Oh, that’s just something those crazy AGfolk do” and never learn them seriously, even for the category of open subsets of a topological space.

Anyway, what’s a manifold? Well, it’s a topological space, equipped with a sheaf satisfying some conditions.  Kind of like a scheme.  This isn’t surprising, because they’re also an example of a locally ringed space, as are pretty much all other geometric objects of interest to most mathematicians.  If you know much about the Mittag-Leffler problem, you know that sheaves are useful here, and the solution is to use the gluing axiom, so complex analysis has it’s hands in sheaves.  Now, that’s not surprising either, considering its proximity to CAG.

A bit further from AG proper, but related to differential geometry a lot like how complex analysis is to CAG, we have DEs.  Take a manifold, write down a differential equation on it, and what happens? People prove all sorts of local theorems, about existence and uniqueness and the like, then patch.  It’s a sheaf! For any DE, there is a sheaf of solutions, and a global solution exists if you can patch one together to get a global section.  This isn’t even the abstract categorical notion of a sheaf, it’s the “classical” one.

And that’s just the beginning.  Sheaf theory is EVERYWHERE, and it’s a shame that so few people outside of AG really pick it up.  So much of mathematics is of the form “if [local condition everywhere] then [global condition]” and the proofs are so often by patching and checking compatibility, sheaves are implicit, but forgotten.

Now, category theory began in algebraic topology, and has come to dominate, as a very useful language and bag of tricks, many areas of mathematics.  It’s time that sheaf theory gets the same treatment properly, and becomes a standard part of the toolkit for everyone who does anything with local-global properties.  And once that happens, I’m sure they’ll start popping up in surprising places.

Back to blogging for a bit, though likely infrequently.  Doing a new series that might count as AG from the beginning, so I’ll put it up there once I’ve got a couple done.  We’re going to start doing some Hodge Theory.